Conformal field theory and the web of quantum chaos diagnostics
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Springer
Received: November 9, 2019 Accepted: January 10, 2020 Published: January 28, 2020
Jonah Kudler-Flam,a Laimei Niea and Shinsei Ryua,b a
Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, IL 60637, U.S.A. b James Franck Institute, University of Chicago, Chicago, Illinois 60637, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: We study three prominent diagnostics of chaos and scrambling in the context of two-dimensional conformal field theory: the spectral form factor, out-of-time-ordered correlators, and unitary operator entanglement. With the observation that all three quantities may be obtained by different analytic continuations of the torus partition function, we address the connections and distinctions between the information that each quantity provides us. In this process, we study the emergence of irrationality from “large-N” limits of rational conformal field theories (RCFTs) as well as the explicit breakdown of rationality for theories with central charges greater than the number of their conserved currents. Our analysis begins to elucidate the intermediate dynamical behavior of theories that bridge the gap between integrable RCFTs and maximally chaotic holographic CFTs. Keywords: Conformal Field Theory, AdS-CFT Correspondence, Conformal and W Symmetry ArXiv ePrint: 1910.14575
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)175
JHEP01(2020)175
Conformal field theory and the web of quantum chaos diagnostics
Contents 1 Introduction & background 1.1 Unification through the torus partition function 1.2 Summary of results
1 6 9 10 10 10 11
3 RCFT 3.1 Minimal models 3.1.1 SFF 3.1.2 OTOC 3.1.3 TOMI 3.2 su(2)k 3.2.1 SFF 3.2.2 OTOC 3.2.3 TOMI
12 14 15 16 17 18 18 19 20
4 Irrational CFT (c < ∞) 4.1 Compactified boson 4.1.1 SFF 4.1.2 OTOC 4.1.3 TOMI
21 22 23 23 24
5 Holographic CFT 5.1 SFF 5.2 OTOC 5.3 TOMI
24 26 26 27
6 Discussion
28
1
Introduction & background
Many-body quantum chaos has garnered immense attention in recent years from a variety of fields due to its key role in the understanding of the emergence of thermal physics [1, 2] and the emergence of Einstein gravity [3–5]. Given this attention and the inherent complexity of characterizing quantum chaos, a slew of diagnostics have been proposed, though there is no
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JHEP01(2020)175
2 Free theories 2.1 SFF 2.2 OTOC 2.3 TOMI
Spectral form factor. Spectral features of a Hamiltonian are good indicators of the ergodic nature of a system. While the level spacing of a Hamiltonian is certainly revealing, the spectral form factor contains further information because it probes the level statistics of both close and far-separated eigenvalues. The spectral form factor is defined as X g(β, t) ≡ |Z(β + it)|2 = e−β(Em +En ) ei(Em −En )t (1.1) n,m
where the inverse temperature β is present as a regulator to cut off the high-energy modes. Later “times” probe correlations between eigenvalues that are closer. For systems with discrete spec
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